A nonsmooth static frictionless contact problem with locking materials

2018 ◽  
Vol 16 (06) ◽  
pp. 851-874 ◽  
Author(s):  
Mircea Sofonea
Keyword(s):  
Continuous Dependence ◽  
Penalty Method ◽  
Optimization Problem ◽  
Convergence Result ◽  
Normal Displacement ◽  
Unique Weak Solution ◽  
Unilateral Constraints ◽  
Weak Formulation ◽  
Frictionless Contact ◽  
Displacement Constraints

We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interface law which involves unilateral constraints and subdifferential conditions. We describe the model and derive its weak formulation, which is in the form of an elliptic variational–hemivariational inequality for the displacement field. Then, we establish the existence of a unique weak solution to the problem. Next, we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a particular version of the model for which we prove the continuous dependence of the solution on the bounds which govern the locking and the normal displacement constraints, respectively. We apply this convergence result in the study of an optimization problem associated to the contact model.

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit

1996 ◽  
Vol 126 (4) ◽  
pp. 837-865 ◽  
Author(s):  
Fernando Reitich ◽  
H. Mete Soner
Keyword(s):  
Surface Tension ◽  
Formal Analysis ◽  
Local Equilibrium ◽  
Convergence Result ◽  
Normal Velocity ◽  
Infinitely Many Solutions ◽  
Weak Formulation ◽  
Material Interfaces ◽  
Internal Surfaces ◽  
Image Position

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equationHere Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires thatIn case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

A New Adaptive Penalty Method for Constrained Genetic Algorithm and Its Application to Water Distribution Systems

2006 ◽  
Author(s):  
Berge Djebedjian ◽  
Ashraf Yaseen ◽  
Magdy Abou Rayan
Keyword(s):  
Genetic Algorithm ◽  
Penalty Function ◽  
Penalty Method ◽  
Distribution Systems ◽  
Water Distribution ◽  
Optimization Problem ◽  
Water Distribution Systems ◽  
Building Blocks ◽  
Adaptive Penalty ◽  
Adaptive Penalty Method

This paper presents a new adaptive penalty method for genetic algorithms (GA). External penalty functions have been used to convert a constrained optimization problem into an unconstrained problem for GA-based optimization. The success of the genetic algorithm application to the design of water distribution systems depends on the choice of the penalty function. The optimal design of water distribution systems is a constrained non-linear optimization problem. Constraints (for example, the minimum pressure requirements at the nodes) are generally handled within genetic algorithm optimization by introducing a penalty cost function. The optimal solution is found when the pressures at some nodes are close to the minimum required pressure. The goal of an adaptive penalty function is to change the value of the penalty draw-down coefficient during the search allowing exploration of infeasible regions to find optimal building blocks, while preserving the feasibility of the final solution. In this study, a new penalty coefficient strategy is assumed to increase with the total cost at each generation and inversely with the total number of nodes. The application of the computer program to case studies shows that it finds the least cost in a favorable number of function evaluations if not less than that in previous studies and it is computationally much faster when compared with other studies.

A Penalty Optimization Algorithm for Eigenmode Optimization Problem Using Sensitivity Analysis

2014 ◽  
Vol 15 (3) ◽  
pp. 776-796 ◽  
Author(s):  
Zhengfang Zhang ◽  
Weifeng Chen ◽  
Xiaoliang Cheng
Keyword(s):  
Sensitivity Analysis ◽  
Optimization Algorithm ◽  
Penalty Method ◽  
Optimization Problem ◽  
Adjoint Variable Method ◽  
Adjoint Variable ◽  
First Variation ◽  
Quadratic Penalty Method ◽  
The First Variation ◽  
Time Term

AbstractThis paper investigates the eigenmode optimization problem governed by the scalar Helmholtz equation in continuum system in which the computed eigenmode approaches the prescribed eigenmode in the whole domain. The first variation for the eigenmode optimization problem is evaluated by the quadratic penalty method, the adjoint variable method, and the formula based on sensitivity analysis. A penalty optimization algorithm is proposed, in which the density evolution is accomplished by introducing an artificial time term and solving an additional ordinary differential equation. The validity of the presented algorithm is confirmed by numerical results of the first and second eigenmode optimizations in 1Dand 2Dproblems.

Weak formulation of axi-symmetric frictionless contact problems with boundary elements

2005 ◽  
Vol 83 (10-11) ◽  
pp. 836-855 ◽  
Author(s):  
Enrique Graciani ◽  
Vladislav Mantič ◽  
Federico París ◽  
Antonio Blázquez
Keyword(s):  
Boundary Elements ◽  
Contact Problems ◽  
Weak Formulation ◽  
Frictionless Contact

Analytical Target Cascading Based on the Quadratic Exterior Penalty Method for Complex System Design

2012 ◽  
Vol 544 ◽  
pp. 164-169
Author(s):  
Xiao Lin Zhang ◽  
Ping Jiang ◽  
Jun Zheng ◽  
Ying Li
Keyword(s):  
Complex System ◽  
System Design ◽  
Penalty Method ◽  
Optimization Problem ◽  
System Optimization ◽  
Analytical Target Cascading ◽  
Decomposition Structure ◽  
Target Cascading ◽  
Complex System Design ◽  
Weight Coefficients

Analytical Target Cascading (ATC) is a method to partition the optimization of a complex system into a set of subsystem optimizations and a single system optimization according to the structure of the complex system, and coordinate subproblems toward an optimal system design. The constructed new optimization problem owns a hierarchical structure, which better matches the real organization structure of complex system design, so the ATC method provides a promising way to deal with the complex system. For each design problem at a given level, an optimization problem is to minimize the discrepancy between its responses and propagated targets. In ATC, for feasibility of subproblems, the target-response pairs are translated into the relaxation terms in which the weight coefficients is used to represent the relative importance of responses and linking variables matching their corresponding target, and achieve acceptable levels of inconsistency between subproblems when top level targets are unattainable in the hierarchical decomposition structure. Furthermore, weighting coefficients influence convergence efficiency and computational efficiency so that the suitable allocation of weight coefficients is a challenge. This paper adopts the Quadratic Exterior Penalty Method to deal with the weight coefficients that achieve solutions within user-specified acceptable inconsistency tolerances. Meanwhile, the method prototype will be tested on a numerical example and implemented using MATLAB and iSIGHT.

Uniqueness and stability of the solution to a thermoelastic contact problem

1990 ◽  
Vol 1 (4) ◽  
pp. 371-387 ◽  
Author(s):  
Peter Shi ◽  
Meir Shillor
Keyword(s):  
Differential Equation ◽  
Contact Problem ◽  
Continuous Dependence ◽  
Initial Temperature ◽  
Integral Transform ◽  
Coupling Constant ◽  
Frictionless Contact ◽  
Linear Thermoelasticity ◽  
Thermoelastic Contact Problem ◽  
Stability Of The Solution

Uniqueness and continuous dependence on the initial temperature are proved for a onedimensional, quasistatic and frictionless contact problem in linear thermoelasticity. First the problem is reformulated in such a way that it decouples. The resulting problem for the temperature is a nonlinear integro-differential equation. Once the temperature is known the displacement is recovered from an appropriate variational inequality. Uniqueness is proved by considering an integral transform of the temperature. The steady solution is obtained and the asymptotic stability is shown. It turns out that the asymptotic behaviour and the steady state are determined by a relation between the coupling constant a and the initial gap.

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